Causality Causality on the Radio I heard a news article on the radio
recently. It reported a survey which had shown that people who smoke are more
likely to suffer from tooth decay than those who don't. The news presenter
concluded that smoking contributed to tooth decay, another good reason to give
up cigarettes.

But how right was he? The survey would have looked at the rates of tooth
decay among people who smoke and among those who don't. It must have found that
there was a statistically significant positive correlation between how many
fillings people have in their teeth and how many cigarettes they smoke each day.
Given such data, can we conclude that smoking causes tooth decay? Would it not
have been equally valid to conclude that having fillings causes people to smoke?

The news presenter applied his prejudices and drew a conclusion which sounds
reasonable without realising that the converse was also a possible explanation
of the survey results. In fact the real reason behind the correlation was
probably a third factor such as general variations in health awareness among the
people surveyed. Those who care about their good health are more likely to brush
their teeth and not to smoke.

The difference between the possible conclusions from the survey is not just
one of semantics. People listening to the radio might have thought that if they
gave up smoking then they would have less tooth decay. They would be wrong. The
correct way for them to prevent tooth decay is to brush their teeth more often.
In this case the false conclusion is not very dangerous, but such false
conclusions about causal relationships drawn from surveys are common. If it is a
survey which shows a correlation between race and crime rates then the wrong
conclusion could lead to increased racism. In truth it is probably the
consequences of racism, not race, which are the real cause of the crime problem
identified by the survey in the first place.

Causality in Physics If your mind is opened a little by my story of the
survey in the news article, then now would be a good time to ask yourself if you
are drawing the wrong conclusion about causality in physics. Suppose you saw
your child bump into a table and an expensive vase fell off smashing into pieces
on the floor. Would you conclude that her carelessness caused the vase to be
broken? Probably you would. Why would you not conclude that the vase falling off
the table caused her to bump, quite innocently, into the table?

Your response might be that, for one thing, the vase was broken after her
collision with the table so the direction of the causal link is incontestable.
Do the laws of physics support such a stance?

To keep things simple, let's start by considering just classical Newtonian
mechanics. The form which the laws of physics take is crucial to our
understanding of causality. Newton's laws take the form of a set of differential
equations describing the motion of particles under forces that act between them.
If we know the initial positions and velocities of all the particles at an
initial time then their positions are determined at any future time. So does
this form for the laws of physics allow us to justify our concept of causality.
It would seem so because the initial conditions seem to be causing all that
happens in the future.

There is a catch. The laws of physics in this form can be made to work
identically in reverse. If we know the final state of a system we can just as
easily determine its past. Newton's laws do not explain why past events are the
cause of future events.

How about the laws of thermodynamics? If we have a system of many particles
then we can not determine all their positions and velocities exactly. When we
know only some statistical information about them they obey laws which seem not
to be reversible. The second laws of thermodynamics says that entropy must
always increase. Could this be linked to causality?

Indeed, the continual increase of entropy is intimately linked to our
perception of causality. Entropy is a measure of disorder in a system and
defines an arrow of time which can be linked to the psychological arrow. There
is, however, a catch. The second law of thermodynamics is inexplicable in terms
of the underlying laws of physics which, as far as we know, are reversible. This
is enshrined in a theorem of relativistic quantum field theory which proves the
necessity of CPT conservation.

The increase of entropy can be understood in certain idealised experiments.
For example, if we take two closed containers filled with gases which are each
in thermal and chemical equilibrium, and allow them to mix by connecting the two
systems without allowing any energy to escape or enter, then when the system
comes back into equilibrium the entropy of the final state can be shown
theoretically to be higher than the combined entropies in the two original
systems. This seems to be theoretical evidence for increasing entropy and it is
confirmed by experiment, but we must not be missled. The assumption that systems
tend towards equilibrium has been justified. We are victims of our prejudices
about causality again and have devised an argument with circular reasoning to
support it.

Physicists have devised many other arguments for why entropy always
increases, trying to get round the problem of CPT symmetry. Here are a few
possibilities:

CPT symmetry exchanges matter for antimatter so perhaps entropy would
decrease for antimatter. Fault: Electromagnetic radiation can not be
distinguished from its antimatter image, and yet it obeys the second law of
thermodynamics. CPT symmetry does not apply to the collapse of the wavefunction
in quantum mechanics which is a time asymmetric process. Query: Does this mean
that the third law of thermodynamics is not valid for classical statistical
mechanics? CPT conservation is violated by quantum gravity This could be true
but can the laws of thermodynamics be a result of quantum gravity who's effects
are normally thought to be irrelevant except in the most extreme physical
regimes. Entropy increases on account of the fact that it started very low at
the beginning of time. Thus it is due to the initial conditions being set in a
special way. and from then on it could only increase. But then why were initial
conditions set rather than final or mixed boundary conditions. When I was an
undergraduate I naively thought that physicists understood entropy. Some have
produced arguments based on any or all of the above possibilities. In retrospect
I think now that I should be no more convinced by any of those arguments than I
should if I heard someone arguing that smoking causes tooth decay based on the
correlation reported in the survey.

One of the difficulties is that we don't really have an ideal definition of
entropy. We can understand it as a measure of disorder in a closed equilibrium
system. More generally we have to resort to some kind of coarse graining process
in which we imagine that a non-equilibrium system can be seen as made of small
sub-systems, or grains, which are in equilibrium themselves but not in
equilibrium with each other.

Entropy might be better understood in terms of information. It can be linked
to the number of bits which are needed to describe a system accurately. In a hot
disordered system you need to specify the individual state of each particle,
while a cold lattice can be described in terms of its lattice shape, size and
orientation. Far less information is needed for the low entropy system.

The claim that entropy increases because it started low in the big bang is
perhaps the one which has fallen into conventional wisdom, even if it is
admitted that we don't understand why it started low. Perhaps it is because of
some huge unknown symmetry which was valid at the high temperatures of the big
bang and broken later. It is not really clear why it should increase all the
time either. Why can't it just go up and down?

In a completely deterministic system the evolution of the system is equally
well determined by its final state as by its initial so we could argue that the
amount of information in the system must be constant. The difficulty there is
that we are assuming an exact knowledge of state which is impossible. In any
case, quantum mechanics is not deterministic. If we make a perfect crystal with
an unstable isotope, as time passes some of the atoms will decay. The amount of
information needed to track the decayed atoms increases. Perhaps, then, it
really is quantum mechanics and the collapse of the wave function which is
responsible.

If physicists used to think they understood entropy then their faith was
deeply shaken when Hawking and Bekenstein discovered that the laws of
thermodynamics could be extended to the quantum mechanics of black holes. The
entropy is given by the area of the black hole but its temperature can only be
understood through quantum mechanical effects. This shows that classical
understanding of thermodynamics is indeed incomplete and perhaps only a complete
theory of quantum gravity can explain the laws fully.

We might try to understand the quantum state of the entire universe by using
Feynman's path integral formulation of quantum mechanics. We must form a sum
over all possible space-time manifolds allowed in general relativity. Hawking
has argued that we can understand entropy in this way if the universe is an
entirely closed system, bounded in both time and space. He claims that there are
two possible ways a universe could start or end. One has low entropy the other
high. The only consistent picture is one in which it is low at one end and high
at the other hence temporal symmetry is broken.

If this argument could be made solid then it would be a powerful one. The
path integral formulation avoids problems of time since it is a sum over all
possible universes rather than an evolutions with separate boundary conditions.
The argument can only be made complete when we understand quantum gravity
better.

At this point our belief in causality seems to rest in our faith that the
universe has a simple topology as described by standard big bang models. This
rests on little more than some limited observations and an application of
Occam's razor. Our measurements of the cosmic microwave backgrounds show a high
degree of isotropy and the universe seems to be homogenous on large scales in so
far as we can tell. Our observation is limited by a horizon defined by the age
of the universe and the speed of light. Thus we cannot observe anything beyond
about 15 billion light years distance. Why should we imagine that the size of
the universe is a similar order of magnitude to its current age? We have been
unable to measure the extent to which space is curved and can not place limits
on its size, or even be sure it is finite.

It seems to be only an application of Occam's razor which justifies the
assumption that space is homogenous on scales hundreds of orders of magnitude
larger than the observable horizon. It is quite possible, as far as we can tell,
that the big bang is actually just a huge white hole which formed in a larger
universe. Perhaps on some huge scale there are a population of black and white
holes of vastly different sizes. What does that say about the laws of
thermodynamics?

Apart from entropy there are other aspects of causality. We know that in
general relativity causal effects are limited by the light cones which are part
of the geometry of space-time. But the geometry is itself dynamic. In general
relativity it is possible to construct space-time models which have closed
time-like paths. If such things really exist in the universe we would be able to
travel back to our past.

Traditionally physicists have simply said that such universes must be ruled
out because if we could travel back to our past we could change our history,
which seems to raise contradictions. Recently some physicists have started to
question this assumption. It seems possible that quantum mechanics may allow
closed time like curves through space-time wormholes to be constructed, at least
in principle. The contradictions which were thought to be a consequence of time
travel do not stand up to close examination.

Perhaps it would be possible to travel back to the past and see our parents
but some chance event would prevent us from being able to change their lives in
ways which we know never happened. If that is a correct interpretation then it
attacks our faith in our own free will.

There is perhaps little that we can conclude reliably about causality from
our current understanding of physics. Only when we have found and understood the
correct theory for quantum gravity will we be able to know the truth. We may be
prevented from finding that theory if we hold fast to our prejudices.

Occam's razor does not have a very good track record in cosmology. Usually
space turns out to contain more complexity than we imagined before we looked. It
will be billions of years before we are able to see beyond the current horizon
defined by the speed of light . In the nearer term theory is our only hope to
know what the structure of space-time is like on very large scales.

Related Reading Cosmology, Time's Arrow, and that old double standard, Huw
Price Nonlocality as an Explanation for Finetuning and Field Replication in
Nature., Bennet, Froggatt and Nielsen Time Machines and the principle of least
action, Carlini, Frolov, Mensky, Novikov, Soleng